Guaranteed error bounds in homogenisation: an optimum stochastic approach to preserve the numerical separation of scales

Paladim, D. A., Moitinho de Almeida, J. P., Bordas, S. P. A. ORCID: https://orcid.org/0000-0001-8634-7002 and Kerfriden, P. ORCID: https://orcid.org/0000-0002-7749-3996 2017. Guaranteed error bounds in homogenisation: an optimum stochastic approach to preserve the numerical separation of scales. International Journal for Numerical Methods in Engineering 110 (2) , pp. 103-132. 10.1002/nme.5348

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Abstract

This paper proposes a new methodology to guarantee the accuracy of the homogenisation schemes that are traditionally employed to approximate the solution of PDEs with random, fast evolving diffusion coefficients. More precisely, in the context of linear elliptic diffusion problems in randomly packed particulate composites, we develop an approach to strictly bound the error in the expectation and second moment of quantities of interest, without ever solving the fine-scale, intractable stochastic problem. The most attractive feature of our approach is that the error bounds are computed without any integration of the fine-scale features. Our computations are purely macroscopic, deterministic and remain tractable even for small scale ratios. The second contribution of the paper is an alternative derivation of modelling error bounds through the Prager–Synge hypercircle theorem. We show that this approach allows us to fully characterise and optimally tighten the interval in which predicted quantities of interest are guaranteed to lie. We interpret our optimum result as an extension of Reuss–Voigt approaches, which are classically used to estimate the homogenised diffusion coefficients of composites, to the estimation of macroscopic engineering quantities of interest. Finally, we make use of these derivations to obtain an efficient procedure for multiscale model verification and adaptation.

Item Type:	Article
Date Type:	Publication
Status:	Published
Schools:	Engineering Advanced Research Computing @ Cardiff (ARCCA)
Publisher:	Wiley
ISSN:	0029-5981
Date of First Compliant Deposit:	15 August 2017
Date of Acceptance:	1 August 2016
Last Modified:	27 Nov 2024 10:15
URI:	https://orca.cardiff.ac.uk/id/eprint/102936

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